The mechanical characteristics of granular matter tested under biaxial loading are studied by numerical simulation. The stress-strain process, volume strain, coordination number, and shear zone are analyzed. The results indicate that the shear zone develops along with the coordination number and volume strain. With an increase in strain, the shear zone is centralized gradually. Moreover, the force transfer in the granular matter is also discussed.Выполнено численное моделирование механических характеристик гранулята, испытуемого в условиях двухосного нагружения. Анализируются зависимость деформации от напряжения, объемная деформация, координационное число и область пластического сдвига. Результаты показывают, что с ростом координационного числа и объемной деформации расширяется область пластического сдвига. С увеличением деформации область пластического сдвига постепенно смещается к центру. Кроме того, обсуждена передача усилия в грануляте

Bi, Z.W.

Huang, W.F.

Xiao, G.X.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

UDC 539.4Simulation of the Mechanical Characteristics of a Granular Matter Testedunder Biaxial LoadingZ. W. Bi,a,b,c,1W. F. Huang,aand G. X. Xiaoaa Resource and Environmental Engineering, Jiangxi University of Science and Technology, Ganzhou,Jiangxi, Chinab Mine Safety Technology Branch, Chinese Coal Research Institute, Beijing, Chinac State Key Laboratory of Coal Mining and Clean Utilization, Beijing, China1 bizongwei1976@163.comThe mechanical characteristics of granular matter tested under biaxial loading are studied bynumerical simulation. The stress-strain process, volume strain, coordination number, and shear zoneare analyzed. The results indicate that the shear zone develops along with the coordination numberand volume strain. With an increase in strain, the shear zone is centralized gradually. Moreover, theforce transfer in the granular matter is also discussed.Keywords: granular matter, force chain, biaxial test simulations, micromechanical.Introduction. The stress–strain response of granular materials shows an incrementallynonlinear behavior when they are subjected to external loading. The macromechanicalbehavior stems not only from phenomena occurring at the contact surface, but also frommesoscale arrangements such as fabric evolution and force chains [1].Numerical simulations using the discrete element method (DEM) have become avaluable tool in the study on different mechanical phenomena occurring at the micro- andmesoscale in granular materials [2]. The simulation of two-dimensional biaxial compressiontest has been performed on a densely packed assembly of polydisperse circular disks. Acontact model that accommodates normal motions is based on the Hook law, while themodel with tangential motion combines the Hook law with the Coulomb law to study thefriction. A more detailed description of the contact laws, specimen preparation, andsimulation parameters are given in [3]. Particle radii are uniformly distributed, andgravitational effects are neglected.To create some initial configuration, non-contacting particles are distributed randomlywithin a box when the confining stress of 2 MPa is applied to all four walls of the box. Theassembly is then allowed to settle to a state where the kinetic energy is negligible. Thisisotropic assembly is subjected to boundary driven biaxial compression. The assemblyundergoes compression at a constant strain rate in the vertical direction, while allowed toexpand in the horizontal direction under constant confining stress. Figure 1 shows theboundary conditions and initial configuration, and Table 1 provides the material propertiesand parameters used. The compression rate in the vertical direction is 0.001 m/s.The stresses are calculated considering that action of stresses on each set of oppositewalls, where stresses on each wall are calculated by dividing the total force acting on thewall by the corresponding specimen length. The axial deviatoric stress is defined as   a c , (1)where  a is the axial stress and  c is the mean confining stress.The strains in both the vertical direction and the horizontal direction are calculatedusing the Eq. (2):© Z. W. BI, W. F. HUANG, G. X. XIAO, 2015218 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1L LL L00 2( ), (2)where L is the specimen current length in the corresponding direction and L0 is thespecimen original length in that direction. The specimen current height and width are usedin stress and strain calculations.The volumetric strain is calculated using the Eq. (3):  V x y  . (3)Macromechanical Observations. The bulk mechanical behavior of the virtualspecimen is represented by stress–strain curve as shown in Fig. 2. The peak or maximumdeviatoric stress attained using the virtual specimen is approximately 3 MPa with the axialstrain of approximately 2.5%. Post peak behavior is observed as strain softening.The strain evolution in the system exhibits distinct regimes of deformation, and theessential aspects of deformation at different stages during loading can be described asfollows. Typically, dense assemblies of cohesive granular materials are compressed in aperiod of strain-hardening regime, followed by strain-softening and, finally, the so-calledresidual stress state [4–6]. The strain hardening regime is characterized by a monotonicincrease with axial strain to a peak value. During the post-peak phase the deviatoric stressdrops significantly along with progressive development of shear bands until it reaches asteady state value called the residual stress state. Relatively large fluctuations are observedat the post peak stage. The ensuing strain-softening regime is characterized by a sequenceof precipitous drops. A single persistent shear band governs the dynamics in the residualstress state regime.T a b l e 1Granular Material ParametersLengthwidth(mm)dmax ,mmdmin ,mmFrictioncoefficientContactstiffness (N)Density(kg/m3)120 60 6.3 4.6 0.5 5 108 1000Fig. 1. Simulation model. Fig. 2. Stress–strain curve.ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1 219Simulation of the Mechanical Characteristics of a Granular Matter ...Volumetric Strain. The total volumetric strain undergoes compression, dilation orincrease of volume, and then continuing deformation, as it is shown in Fig. 3. The suddendecrease of dilation rate coincides with the onset of shear band. The maximum rate ofdilation coincides with the peak strength.As it can be seen under compression the value V decreases to be negative. Whilethe stress increases, the system displays elastic strain at this time. When  	 2%,V 08. %, the system moves towards the most dense state (point A) with the maximumvolume fraction. When the value  2%, V starts to increase, the system is in the plasticstage. Until  35. % and V  0, the volumetric strain returns to its initial value (point B).After  35. %, the rate of change (slope) remains the same, even though there is slightfluctuation in the volumetric strain. When  4 2. %, V continues to increase, but the rateis significantly lower (point C). Hereafter, the system reaches its critical state where theinternal force chain collapses and re-combines constantly, and larger fluctuations of Vappear, respectively.Shear Bands. Some analysis has been conducted on the displacement of shear zone[7–12]. For instance, Vardoulakis and Aifantis [7] and Chambon et al. [8] studied the shearband based on the second gradient method. Meanwhile, Cundall and Strack [2], Bardet andProubet [9] examined shear strain localization by using PFC discrete element software.This paper is focused on the formation and development of shear band considering therotation of particles. In Fig. 4, white spots represent the particles with rotation angle  5 ,grey represents 5 10   , and dark represents  10 . As it can be seen, in the final statewith  15%, most particles with large rotation angle are found to be concentrated insidethe shear band.As it is shown in Fig. 4 the formation of shear bands is accompanied by particlerotation. When  2.2%, plastic deformation takes place in the system, which changesfrom the densest to loose. At this time, some small shear bands formed by the particles withlarger rotation angles are observed, and the band width is similar in value to single particlediameter. It is indicated that localization occurs inside the system before the stress reachesthe peak. These small shear bands are parallel to each other and extend along their owndirection. When  increases, the shear bands start to combine and change their directionnear the combination point. The shear band in one direction becomes more pronounced,while those in another direction start to disappear. In other words, the dominant shear zonestarts to form. With an increase of  the multiple shear band formation, combination, andFig. 3. Volumetric strain curve.220 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1Z. W. Bi, W. F. Huang, and G. X. Xiaobifurcation can be observed. At  4%, more particles have larger rotation angle, and thedistribution is concentrated. A distinct shear zone of certain thickness crosses the entiresystem indicating the system critical state. With further increase of  the shear band formseventually, whose width is of about 6–8 particle diameter. Meanwhile, small shear bandsare observed to disappear. The critical state and the direction of shear band indicate that thegranular system is consistent with the Mohr–Coulomb standards. Similar phenomenon isalso observed during testing on dry sand using scanning electron microscope, digital opticalimage analyzer, and X-ray equipment.Coordination Number. The simplest way to study the microstructural arrangement ofgranular sample is to examine the evolution of coordination number Z. The coordinationnumber Z is defined as mean number of contacts per particle. To observe the change of Z,three measurement balls are placed in the system, as it is shown in Fig. 5a. The evolution ofcoordination number is shown in Fig. 5b.At low strain values there is an insignificant increase of the coordination number withthe increase of strain. And the amount of increase is different with various measurementa b cFig. 4. Distribution of particle rotation angle: (a)   2.2%; (b)   4%; (c)  15%.a bFig. 5. Positions of three measurement balls (a) and evolution of coordination number (b).ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1 221Simulation of the Mechanical Characteristics of a Granular Matter ...balls. When the strain is 1% the coordination number of ball 2 shows a peak value of 3.8,then it decreases. Meanwhile, the coordination numbers of balls 1 and 3 remain at 4.4 and4.0, respectively. When the strain is 2.8%, all particles are without contact, and the systemis at the strain softening stage. When the strain is 4.5%, the system has reached the residualstress state. The coordination numbers of balls 1 and 3 are observed to change only within asmall range, which indicates that the system has completed the transfer from X shear zoneto the sole shear zone.Development of Force Chain. Figure 6 shows the development of force chain atdifferent stages. At  0, the granular matter is in isotropic compression state. A ring-likestructure is observed in most of force chains, their spatial distribution is relatively uniform(Fig. 6a). There is no obvious inhomogeneity of force chain intensity, which means the loadat the boundary/bounds is distributed evenly on most particles.If the load is imposed on the upper and lower boundaries, the major direction ofprincipal stress is vertical, while the minor is horizontal. The force chain structure ischanged to column parallel to the vertical direction of major principal stress (Fig. 6b). Atthis time, the intensity of force chain shows some inhomogeneity. The strong force chainsbear the load in the vertical direction. Although the weak ones do not share the verticalload, they play an important role in supporting the vertical force chains and preventingthem from bending.When the system experiences the stress dilatancy phase with  2.8%, the local voidratio increases, and the force chains become annular (or spiral) and form the so-called arch.Due to reduction of contact number, the strong force chains lose the support of weak forcechains, and the particles rotate significantly. The annular force chain is very unstable andtends to fracture resulting in a whirlpool in the local area. When the force chain is broken,microcracks appear. The development of multiple microcracks leads to the formation ofshear bands. With the dominant shear band the force chains outside the shear band remainto be a column structure parallel to the vertical direction, while the ones inside the shearband undergo significant bending. It is indicated that the principle stress inside the shearband has rotated (Fig. 6c). The bent force chains inside the shear band are ring-like, andtheir width is consistent with the width of shear band. After the formation of shear band,the direction of particle contact angle also changes, since some force chains inside the shearband are not parallel to the principal stress, however, are at certain angle.Fig. 6. Development of column-like structure and its fracture in shear bands (scale = 3 106 ).a b c222 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1Z. W. Bi, W. F. Huang, and G. X. XiaoConclusions. In this paper, the relation between the inter-particle factors and themacromechanical behavior of granular materials was investigated. The results show that thevariation of stress has a significant effect on the evolution of displacement, coordinationnumber and force chain in the granular matter. With an increase in strain, the shear zonebecomes more evident. An overall translation phenomenon is observed when the strain isover 4.8%. Furthermore, it is found that the system at residual stress state develops forcechains that are highly susceptible to fracture, which leads to strong stress fluctuations.Stress collapses also reduce the contacts, and therefore result in a temporary stability of thegranular specimen.Acknowledgments. This work has been supported by Chinese Postdoctoral ScienceFoundation (2014M550358).1. F. Darve, “The expression of rheological laws in incremental form and the mainclasses of constitutive equations,” in: F. Darve (Ed.), Fractured and Jointed RockMasses, Taylor & Francis, London (1990), pp. 123–148.2. P. A. Cundall and O. D. L. Strack, “A discrete numerical model for granularassemblies,” Geotechnique, 29, No. 1, 47–65 (1979).3. Q. C. Sun and F. Jin, “Force chain in a uniaxially compressed static granular matter in2D,” Acta Phys. Sinica, 58, No. 9, 6549–6553 (2009).4. A. A. Pena and A. F. Lizcano, “Biaxial test simulations using a packing of polygonalparticles,” Int. J. Num. Anal. Meth. Geomech., 32, No. 2, 143–160 (2008).5. A. Tordesillas, “Force chain buckling, unjamming transitions and shear banding indense granular assemblies,” Phil. Mag., 87, No. 32, 4987–5016 (2010).6. S. Luding, M. Latzel, W. Volk, et al., “Discrete element simulations to a continuummodel,” Comp. Meth. Appl. Mech. Eng., 191, No. 1, 21–28 (2001).7. I. Vardoulakis and E. C. Aifantis, “A gradient flow theory of plasticity for granularmaterials,” Acta Mech., 87, No. 3, 197–217 (1991).8. G. Chambon, J. Schmittbuhl, A. Corfdir, et al., “Shear with comminution of a granularmaterial: microscopic deformations outside the shear band,” Phys. Rev. E, 68, 011304(2003).9. J. P. Bardet and J. Proubet, “Numerical simulations of shear bands in idealizedgranular materials,” Solid State Phenom., 23-24, No. 1, 473–481 (1992).10. J. Desrues and G. Viggiani, “Strain localization in sand: an overview of theexperimental results obtained in Grenoble using stereophotogrammetry,” Int. J. Num.Anal. Meth. Geomech., 28, No. 4, 279–321 (2004).11. W. J. Wang, X. Z. Kong, and Z. G. Zhu, “Friction and relative energy dissipation insheared granular materials,” Phys. Rev. E, 75, 041302 (2007).12. M. Oda, K. Iwashita, and H. Kazama, “Micro-structure developed in shear bands ofdense granular soils and its computer simulation – mechanism of dilatancy andfailure,” in: N. A Fleck and A. C. E. Cocks (Eds.), IUTAM Symp. on Mechanics ofGranular and Porous Materials, Kluwer, Dordrecht (1997), pp. 353–364.Received 20. 10. 2014ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1 223Simulation of the Mechanical Characteristics of a Granular Matter ...

Simulation of the Mechanical Characteristics of a Granular Matter Tested under Biaxial Loading

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Simulation of the Mechanical Characteristics of a Granular Matter Tested under Biaxial Loading

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